# Mathematical Art of M C Esher

Susan van Zyl 642 posts |
Maurits Cornelis Escher, who was born in Leeuwarden, Holland in 1898, created unique and fascinating works of art that explore and exhibit a wide range of mathematical ideas. TESSELLATIONS Whether or not this is fair to the mathematicians, it is true that they had shown that of all the regular polygons, only the triangle, square, and hexagon can be used for a tessellation. (Many more irregular polygons tile the plane – in particular there are many tessellations using irregular pentagons.) Escher exploited these basic patterns in his tessellations, applying what geometers would call reflections, glide reflections, translations, and rotations to obtain a greater variety of patterns. He also elaborated these patterns by “distorting” the basic shapes to render them into animals, birds, and other figures. These distortions had to obey the three, four, or six-fold symmetry of the underlying pattern in order to preserve the tessellation. The effect can be both startling and beautiful. Regular Division In Reptiles the tessellating creatures playfully escape from the prison of two dimensions and go snorting about the destop, only to collapse back into the pattern again. Escher used this reptile pattern in many hexagonal tessellations. In Development 1, it is possible to trace the developing distortions of the square tessellation that lead to the final pattern at the center. POLYHEDRA
THE SHAPE OF SPACE
All of Escher’s works reward a prolonged stare, but this one does especially. Somehow, Escher has turned space back into itself, so that the young man is both inside the picture and outside of it simultaneously. The secret of its making can be rendered somewhat less obscure by examining the grid-paper sketch the artist made in preparation for this lithograph. Note how the scale of the grid grows continuously in a clockwise direction. And note especially what this trick entails: A hole in the middle. A mathematician would call this a singularity, a place where the fabric of the space no longer holds together. There is just no way to knit this bizarre space into a seamless whole, and Escher, rather than try to obscure it in some way, has put his trademark initials smack in the center of it. THE LOGIC OF SPACE Cube withRibbons High and Low A third type of “impossible drawing” relies on the brain’s insistence upon using visual clues to construct a three-dimensional object from a two-dimensional representation, and Escher created many works which address this type of anomaly. SELF-REFERENCE AND INFORMATION |

Sally Sargent 498 posts |
This is a very well written and complete article on tessellations. When I was teaching high school art there was another teacher who was a math teacher as well as an art teacher. She introduced tessellations to the students which was a perfect lead in to further studies on Escher. Her approach was pretty simplistic. Your article rounded out my understanding of the math connection to Escher’s art. Thank you so much for this very informative article. sally sargent |

Susan van Zyl 642 posts |
Thanx Sally |